Publications on Crystals, Soft Matter and Phase Transitions, by Charles Radin

  1. C. S. Gardner and C. Radin, The infinite volume ground state of the Lennard-Jones potential, J. Stat. Phys., 20(1979), 719-724. print.pdf; (higher resolution)
  2. G. Hamrick and C. Radin, The symmetry of ground states under perturbation, J. Stat. Phys., 21(1979), 601-607. print.pdf; (higher resolution)
  3. R. Heitmann and C. Radin, The ground state for sticky disks, J. Stat. Phys., 22(1980), 281-287. print.pdf; (higher resolution)
  4. B. Borden and C. Radin, The crystal structure of the noble gases, J. Chem. Phys., 75(1981), 2012-2013. print.pdf
  5. C. Radin, The ground state for soft disks, J. Stat. Phys., 26(1981), 365-373. print.pdf; (higher resolution)
  6. C. Radin, Crystalline symmetry and surface tension, Physica, 113A(1982), 338-342. print.pdf
  7. F. Nicolò and C. Radin, A first-order phase transition between crystal phases in the shift model, J. Stat. Phys., 28(1982), 473-478. print.pdf; (higher resolution)
  8. C. Radin and L. Schulman, Periodicity of classical ground states, Phys. Rev. Lett., 51(1983), 621-622. print.pdf
  9. C. Radin, Classical ground states in one dimension, J. Stat. Phys., 35(1984), 109-117. print.pdf; (higher resolution)
  10. C. Radin, Tiling, periodicity, and crystals, J. Math. Phys. 26(1985), 1342-1344. print.pdf
  11. C. Radin, Crystals and quasicrystals: a lattice gas model, Phys. Lett., 114A(1986), 381-383. print.pdf
  12. C. Radin, Crystals and quasicrystals: a continuum model, Comm. Math. Phys., 105(1986), 385-390. print.pdf
  13. C. Radin, Correlations in classical ground states, J. Stat. Phys., 43(1986), 707-712. print.pdf; (higher resolution)
  14. J. Miekisz and C. Radin, The unstable chemical structure of quasicrystalline alloys, Phys. Lett., 119A(1986), 133-134. print.pdf
  15. J. Miekisz and C. Radin, The third law of thermodynamics, Mod. Phys. Lett., B1(1987), 61-66. print.pdf; (higher resolution)
  16. C. Radin, Low temperature and the origin of crystalline symmetry, Intl. J. Mod. Phys., B1(1987), 1157-1191. print.pdf; (higher resolution)
  17. J. Miekisz and C. Radin, Why solids are not really crystalline, Phys. Rev., B39(1989), 1950-1952. print.pdf
  18. C. Radin, Ordering in lattice gases at low temperature, J. Phys., A22(1989), 317-319. print.pdf
  19. C. Gardner, J. Miekisz, C. Radin and A. Van Enter), Fractal ground state for an Ising model, J. Phys. A22(1989), L1019-L1023. print.pdf
  20. C. Radin, Disordered Ground States of Classical Lattice Models, Revs. Math. Phys. 3(1991), 125-135. print.pdf; (higher resolution)
  21. C. Radin, Global Order from Local Sources, Review-Expository Paper, Bull. Amer. Math. Soc. 25(1991), 335-364. print.pdf; (higher resolution)
  22. D. Berend and C. Radin, Are there chaotic tilings?, Comm. Math. Phys. 152(1993), 215-219. print.pdf; Original, longer version: preprint.pdf
  23. C. Radin, Quasicrystals and geometry, Notices Amer. Math. Soc. 43(1996), 416-421. print.ps;  print.pdf
  24. H. Koch, C. Radin and R. de la Llave, Aubry-Mather theory for functions on lattices, Discrete Contin. Dynam. Systems 3(1997), 135-151. preprint.ps;  preprint.pdf
  25. C. Radin, Rashomon: pavages et rotations, Gaz. Soc. Math. de France 79(1999), 49-57. print.pdf
  26. C. Radin, Statistical Symmetry, chapter in "Geometry at Work", MAA Notes volume 53 (Math. Assoc. Amer., Washington, DC, 2000). print.pdf; (higher resolution)
  27. C. Radin, Symmetries of quasicrystals, J. Stat. Phys. 95(1999), 827-833. print.ps;  print.pdf
  28. C. Radin, Existence of ground state configurations, Math. Phys. Electron. J. 10(2004), Paper 6. (electronic) print.ps;  print.pdf
  29. C. Radin, A review of two books on sphere packing, Amer. Math. Monthly 113(2006), 87-90. print.pdf
  30. C. Radin and L. Sadun, Structure of the hard sphere solid, Phys. Rev. Lett. 94(2005), 015502. print.ps;  print.pdf
  31. H. Koch, C. Radin and L. Sadun, Most stable structure for hard spheres, Phys. Rev. E 72(2005), 016708. print.ps;  print.pdf
  32. L. Bowen, R. Lyons, C. Radin and P. Winkler, Fluid-solid transition in a hard-core system, Phys. Rev. Lett. 96(2006), 025701. print.ps;  print.pdf
  33. L. Bowen, R. Lyons, C. Radin and P. Winkler, A solidification phenomenon in random packings, SIAM J. Math. Anal. 38(2006), 1075-1089. print.pdf
  34. M. Schröter, S. Nägle, C. Radin and H.L. Swinney, Phase transition in a static granular system, EPL 78 (2007), 44004. print.pdf
  35. C. Radin, Conference report II, Appl. Rheol. 17:3(2007), 166. print.pdf,
  36. C. Radin, Random close packing of granular matter, J. Stat. Phys. 131(2008), 567-573. print.pdf
  37. D. Aristoff and C. Radin, Random loose packing in granular matter, J. Stat. Phys., 135(2009), 1-23. print.pdf
  38. J. Bellissard, C. Radin and S. Shlosman, The characterization of ground states, J. Phys. A: Math. Theor. 43 (2010), 305001. print.pdf
  39. D. Aristoff and C. Radin, Random close packing in a granular model, J. Math. Phys. 51 (2010), 113302. print.pdf
  40. H. Koch and C. Radin, Modelling quasicrystals at positive temperature, J. Stat. Phys., 138(2010), 465-475. print.pdf
  41. D. Aristoff and C. Radin, Layering in crumpled sheets, EPL 91(2010), 56003. print.pdf
  42. D. Aristoff and C. Radin, Dilatancy transition in a granular model, J. Stat. Phys. 143(2011), 215-225. print.pdf
  43. T. Argo, M. Guild, P. Wilson, M. Schröter, C. Radin and H. Swinney, Sound speed in water-saturated glass beads as a function of frequency and porosity, J. Acoust. Soc. Am. 129(2011) EL101-EL107. print.pdf
  44. J.-F. Métayer, D.J. Suntrup III, C. Radin, H.L. Swinney and M. Schröter, Shearing of frictional sphere packings, EPL 93(2011), 64003 preprint.pdf
  45. D. Aristoff and C. Radin, First order phase transition of a long polymer chain, J. Phys. A: Math. Theor. 44(2011), 065004. print.pdf
  46. D. Aristoff and C. Radin, First order phase transition in a model of quasicrystals, J. Phys. A: Math. Theor. 44(2011), 255001. print.pdf
  47. D. Aristoff and C. Radin, Rigidity in solids, J. Stat. Phys. 144(2011) 1247-1255. print.pdf
  48. C. Radin and M. Yin, Phase transitions in exponential random graphs, Ann. Appl. Probab. 23(2013) 2458-2471. print.pdf
  49. D. Aristoff and C. Radin, Emergent structures in large networks, J. Appl. Probab. 50(2013) 883-888. print.pdf
  50. C. Radin, A revolutionary material, Notices Amer. Math. Soc. 60(2013) 310-315. print.pdf
  51. C. Radin and L. Sadun, Phase transitions in a complex network, J. Phys. A: Math. Theor. 46 (2013) 305002 print.pdf;    The previous version, with title `A mean field analysis of the fluid/solid phase transition', is here
  52. C. Radin and L. Sadun, Singularities in the entropy of asymptotically large simple graphs. J. Stat. Phys. 158 (2015) 853-865 print.pdf
  53. C. Radin, K. Ren and L. Sadun, The asymptotics of large constrained graphs, J. Phys. A: Math. Theor. 47 (2014) 175001 print-color.pdf    print-grayscale.pdf
  54. R. Kenyon, C. Radin, K. Ren and L. Sadun, Multipodal structure and phase transitions in large constrained graphs, J. Stat. Phys. (2017) 233-258 doi:10.1007/s10955-017-1804-0 print
  55. R. Kenyon, D. Král’, C. Radin and P. Winkler, Permutations with fixed pattern densities, Random Structures Algorithms, 56(2020) 220-250. print.
  56. R. Kenyon, C. Radin, K. Ren and L. Sadun, Bipodal structure in oversaturated random graphs, Int. Math. Res. Notices 2018(2016) 1009-1044 print
  57. C. Radin, Phases in large combinatorial systems, Ann. Inst. H. Poincaré D 5 (2018) 287-308 print.
  58. C. Radin, K. Ren and L. Sadun, A symmetry breaking transition in the edge/triangle network model, Ann. Inst. H. Poincaré D 5 (2018) 251-286 print.
  59. C. Radin, The open mathematics of crystallization, Notices Amer. Math. Soc. 64(2017) 551-556. print. There is a more technical version: `Wulff shape for equilibrium phases', arXiv:1610.08564v2 , here
  60. R. Kenyon, C. Radin, K. Ren and L. Sadun, The phases of large networks with edge and triangle constraints, J. Phys. A: Math. Theor. 50(2017) 435001. print
  61. F. Rietz, C. Radin, H. Swinney and M. Schröter, Nucleation in sheared granular matter, Phys. Rev. Lett. 120(2018) 055701. print.    Supplemental Material.    Movie from Supplementary Material.
  62. C. Radin, K. Ren and L. Sadun, Surface effects in dense random graphs with sharp edge constraint, arXiv:1709.01036v2 preprint
  63. C. Radin and H. Swinney, Phases of granular matter, J. Stat. Phys. 175(2019) 542-553 doi.org/10.10/s10955-018-2144-4 print
  64. J. Neeman, C. Radin and L. Sadun, Phase transitions in finite random networks, J Stat Phys 181 (2020) 305-328, print.    Title of previous version: `Nucleation during phase transitions in random networks'.    Supplementary movies at https://www.youtube.com/playlist?list=PLZcI2rZdDGQqx3WEY6BXoJXcxqQXF5ZzQ
  65. W. Jin, C. O'Hern, C. Radin, M. Shattuck and H.L. Swinney, Homogeneous crystallization in cyclically sheared frictionless grains, Phys. Rev. Lett. 125 (2020) 258003. print.    Supplemental Material.
  66. C. Radin, Conway and aperiodic tilings, Math. Intell. 43 (2021) 15-20, print.
  67. J. Neeman, C. Radin and L. Sadun, Moderate deviations in cycle count, Random Struct. Alg. 63 (2023) 779-820, print.
  68. C. Radin, Equivalence between random close packing in granular matter and freezing in the hard sphere model, arxiv:2105.09982, preprint.
  69. J. Neeman, C. Radin and L. Sadun, Typical large graphs with given edge and triangle densities, Probab. Theory Relat. Fields (2023), https://doi.org/10.1007/s00440-023-01187-8. arxiv:2110.14052.
  70. J. Neeman, C. Radin and L. Sadun, Existence of a symmetric bipodal phase in the edge-triangle model, J. Phys. A: Math. Theor. 57 (2024) 095003. print. Publications of Charles Radin

    Publications of Charles Radin

    1. C. Radin, Approach to equilibrium in a simple model, J. Math. Phys., 11(1970), 2945-2955. print.pdf
    2. C. Radin, Noncommutative mean ergodic theory, Comm. Math. Phys., 21(1971), 291-302. print.pdf
    3. C. Radin, Gentle perturbations, Comm. Math. Phys., 23(1971), 189-198. print.pdf
    4. G. Emch and C. Radin, Relaxation of local thermal deviations from equilibrium, J. Math. Phys., 12(1971), 2043-2046. print.pdf
    5. C. Radin, Ergodicity in von Neumann algebras, Pac. J. Math., 48(1973), 235-239. print.pdf
    6. C. Radin, Automorphisms of von Neumann algebras as point transformations, Proc. Amer. Math. Soc., 39(1973), 343-346. print.pdf
    7. C. Radin, Dynamics of limit models, Comm. Math. Phys., 33(1973), 283-292. print.pdf
    8. C. Radin, A noncommutative L_1-mean ergodic theorem, Advances in Math., 21(1976), 110-111. print.pdf
    9. C. Radin, Some remarks on the evolution of a Schrödinger particle in an attractive 1/r^2 potential, J. Math. Phys., 16(1975), 544-547. print.pdf
    10. C. Radin, Particle lattice models and the dynamical instability of many-body systems, Comm. Math. Phys., 44(1975), 165-168. print.pdf
    11. R.T. Powers and C. Radin, Average boundary conditions in Cauchy problems, J. Functional Analysis, 23(1976), 23-32. print.pdf
    12. C. Radin, The dynamical instability of nonrelativistic many-body systems, Comm. Math. Phys., 54(1977), 69-80. print.pdf
    13. C. Radin, Pointwise ergodic theory and operator algebras, J. Math. Phys. 19(1978), 1983-1985. print.pdf
    14. C. Radin and B. Simon, Invariant domains for the time-dependent Schrödinger equation, J. Diff. Eqns. 29(1978), 289-296. print.pdf
    15. J. A. Goldstein, C. Radin and R. E. Showalter, Convergence rates of ergodic limits for semigroups and cosine functions, Semigroup Forum, 16(1978), 89-95. print.pdf
    16. C. Radin, Signal propagation in lattice models of quantum many body systems, Comm. Math. Phys., 62(1978), 159-166. print.pdf
    17. C. S. Gardner and C. Radin, The infinite volume ground state of the Lennard-Jones potential, J. Stat. Phys., 20(1979), 719-724. print.pdf; (higher resolution)
    18. G. Hamrick and C. Radin, The symmetry of ground states under perturbation, J. Stat. Phys., 21(1979), 601-607. print.pdf; (higher resolution)
    19. R. Heitmann and C. Radin, The ground state for sticky disks, J. Stat. Phys., 22(1980), 281-287. print.pdf; (higher resolution)
    20. B. Borden and C. Radin, The crystal structure of the noble gases, J. Chem. Phys., 75(1981), 2012-2013. print.pdf
    21. C. Radin, The ground state for soft disks, J. Stat. Phys., 26(1981), 365-373. print.pdf; (higher resolution)
    22. C. Radin, Crystalline symmetry and surface tension, Physica, 113A(1982), 338-342. print.pdf
    23. F. Nicolò and C. Radin, A first-order phase transition between crystal phases in the shift model, J. Stat. Phys., 28(1982), 473-478. print.pdf; (higher resolution)
    24. C. Radin and L. Schulman, Periodicity of classical ground states, Phys. Rev. Lett., 51(1983), 621-622. print.pdf
    25. C. Radin, Classical ground states in one dimension, J. Stat. Phys., 35(1984), 109-117. print.pdf; (higher resolution)
    26. C. Radin, Tiling, periodicity, and crystals, J. Math. Phys. 26(1985), 1342-1344. print.pdf
    27. C. Radin, Crystals and quasicrystals: a lattice gas model, Phys. Lett., 114A(1986), 381-383. print.pdf
    28. C. Radin, Crystals and quasicrystals: a continuum model, Comm. Math. Phys., 105(1986), 385-390. print.pdf
    29. C. Radin, Correlations in classical ground states, J. Stat. Phys., 43(1986), 707-712. print.pdf; (higher resolution)
    30. J. Miekisz and C. Radin, The unstable chemical structure of quasicrystalline alloys, Phys. Lett., 119A(1986), 133-134. print.pdf
    31. J. Miekisz and C. Radin, The third law of thermodynamics, Mod. Phys. Lett., B1(1987), 61-66. print.pdf; (higher resolution)
    32. C. Radin, Low temperature and the origin of crystalline symmetry, Intl. J. Mod. Phys., B1(1987), 1157-1191. print.pdf; (higher resolution)
    33. C. Radin, Symmetry: Unifying Human Understanding, Amer. Scientist, 76(1988), 392-393. print.pdf; (higher resolution)
    34. J. Miekisz and C. Radin, Why solids are not really crystalline, Phys. Rev., B39(1989), 1950-1952. print.pdf
    35. C. Radin, Ordering in lattice gases at low temperature, J. Phys., A22(1989), 317-319. print.pdf
    36. C. Gardner, J. Miekisz, C. Radin and A. Van Enter) Fractal ground state for an Ising model, J. Phys. A22(1989), L1019-L1023. print.pdf
    37. C. Radin, Disordered Ground States of Classical Lattice Models, Revs. Math. Phys. 3(1991), 125-135. print.pdf; (higher resolution)
    38. C. Radin, Global Order from Local Sources, Review-Expository Paper, Bull. Amer. Math. Soc. 25(1991), 335-364. print.pdf; (higher resolution)
    39. C. Radin and M. Wolff, Space tilings and local isomorphism, Geometriae Dedicata 42(1992), 355-360. print.pdf; (higher resolution)
    40. C. Radin, Z^n versus Z actions for systems of finite type, Contemporary Mathematics 135(1992), 339-342. print.pdf; (higher resolution)
    41. D. Berend and C. Radin, Are there chaotic tilings?, Comm. Math. Phys. 152(1993), 215-219. print.pdf; Original, longer version: preprint.pdf
    42. C. Radin, Symmetry of tilings of the plane, Bull. Amer. Math. Soc. 29(1993), 213-217. print.pdf
    43. C. Radin, The pinwheel tilings of the plane, Annals of Math. 139(1994), 661-702. print.ps;  print.pdf
    44. C. Radin, Symmetry and tilings, Notices Amer. Math. Soc. 42(1995), 26-31. print.ps;  print.pdf
    45. C. Radin, Space tilings and substitutions, Geometriae Dedicata 55(1995), 257-264. print.pdf; (higher resolution)
    46. C. Radin, Aperiodic tilings in higher dimensions, Proc. Amer. Math. Soc. 123(1995), 3543-3548. print.ps;  print.pdf
    47. C. Radin, Quasicrystals and geometry, Notices Amer. Math. Soc. 43(1996), 416-421. print.ps;  print.pdf
    48. C. Radin, Miles of Tiles, Ergodic theory of Z^d-actions, London Math. Soc. Lecture Notes Ser. 228(1996), Cambridge Univ Press, 237-258. preprint.ps;  preprint.pdf
    49. C. Radin and L. Sadun, The isoperimetric problem for pinwheel tilings, Comm. Math. Phys. 177(1996), 255-263. print.pdf
    50. H. Koch, R. de la Llave and C. Radin, Aubry-Mather theory for functions on lattices, Discrete Contin. Dynam. Systems 3(1997), 135-151. preprint.ps;  preprint.pdf
    51. C. Radin, Aperiodic tilings, ergodic theory and rotations, The Mathematics of Long-Range Aperiodic Order, Nato Adv. Sci. Institutes Ser. 489(1997), Kluwer, 499-519. print.pdf; (higher resolution)
    52. J.H. Conway and C. Radin, Quaquaversal tilings and rotations, Inventiones math. 132(1998), 179-188. print.ps;  print.pdf
    53. C. Radin and L. Sadun, Subgroups of SO(3) associated with tilings, J. Algebra 202(1998), 611-633. print.ps;  print.pdf
    54. C. Radin, Statistical Symmetry, chapter in "Geometry at Work", MAA Notes volume 53 (Math. Assoc. Amer., Washington, DC, 2000). print.pdf; (higher resolution)
    55. C. Radin and L. Sadun, An algebraic invariant for substitution tiling systems, Geometriae Dedicata, 73(1998), 21-37. print.ps;  print.pdf
    56. C. Radin, Rashomon: pavages et rotations, Gaz. Soc. Math. de France 79(1999), 49-57. print.pdf; (higher resolution)
    57. C. Radin and L. Sadun, On 2-generator subgroups of SO(3), Trans. Amer. Math. Soc. 351(1999), 4469-4480. print.ps;  print.pdf
    58. J.H. Conway, C. Radin and L. Sadun, On angles whose squared trigonometric functions are rational, Discrete Comput. Geom. 22(1999), 321-332. print.ps;  print.pdf
    59. C. Radin, Symmetries of quasicrystals, J. Stat. Phys. 95(1999), 827-833. print.ps;  print.pdf
    60. J.H. Conway, C. Radin and L. Sadun, Relations in SO(3) supported by geodetic angles, Discrete Comput. Geom. 23(2000), 453-463. print.ps;  print.pdf
    61. C. Radin and L. Sadun, Isomorphism of hierarchical structures, Ergodic Theory Dynam. Systems 21(2001), 1239-1248. print.ps;  print.pdf
    62. N. Ormes, C. Radin and L. Sadun, A homeomorphism invariant for substitution tiling spaces, Geometriae Dedicata 90(2002), 153-182. print.ps;  print.pdf
    63. L. Bowen and C. Radin, Densest packing of equal spheres in hyperbolic space, Discrete Comput. Geom. 29(2003), 23-39. print.ps;  print.pdf
    64. L. Bowen and C. Radin, Optimally dense packings of hyperbolic space, Geometriae Dedicata 104(2004), 37-59. print.ps print.pdf
    65. C. Radin, The symmetry of optimally dense packings, in Non-Euclidean Geometries: János Bolyai Memorial Volume, Mathematics and Its Applications, Vol. 581, Springer, 2006, pp. 197-207. print.pdf;  (higher resolution)
    66. L. Bowen, C. Holton, C. Radin and L. Sadun, Uniqueness and symmetry in problems of optimally dense packings, Math. Phys. Electron. J. 11(2005), Paper 1, 34pp. (electronic) print.ps;  print.pdf
    67. C. Radin, Orbits of orbs: sphere packing meets Penrose tilings, Amer. Math. Monthly 111(2004), 137-149. print.ps;  print.pdf
    68. C. Holton, C. Radin and L. Sadun, Conjugacies for tiling dynamical systems, Comm. Math. Phys. 209(2005), 343-359. print.ps;  print.pdf
    69. C. Radin, Existence of ground state configurations, Math. Phys. Electron. J. 10(2004), Paper 6, 7pp. (electronic). print.ps;  print.pdf
    70. C. Holton and C. Radin, Topological conjugacy between aperiodic tiling dynamical systems. preprint.ps;  preprint.pdf
    71. C. Radin and L. Sadun, Structure of the hard sphere solid, Phys. Rev. Lett. 94(2005), 015502. print.ps;  print.pdf
    72. C. Radin, A review of two books on sphere packing, Amer. Math. Monthly 113(2006), 87-90. print.pdf
    73. H. Koch, C. Radin and L. Sadun, Most stable structure for hard spheres, Phys. Rev. E 72(2005), 016708. print.ps;  print.pdf
    74. L. Bowen, R. Lyons, C. Radin and P. Winkler, Fluid-solid transition in a hard-core system, Phys. Rev. Lett. 96(2006), 025701. print.ps;  print.pdf
    75. L. Bowen, R. Lyons, C. Radin and P. Winkler, A solidification phenomenon in random packings, SIAM J. Math. Anal. 38(2006), 1075-1089. print.pdf
    76. M. Schröter, S. Nägle, C. Radin and H.L. Swinney, Phase transition in a static granular system, EPL 78(2007), 44004. print.pdf
    77. C. Radin, Conference report II, Appl. Rheol. 17:3(2007), 166.
    78. C. Radin, Random close packing of granular matter, J. Stat. Phys. 131(2008), 567-573. print.pdf
    79. D. Aristoff and C. Radin, Random loose packing in granular matter, J. Stat. Phys., 135(2009), 1-23. print.pdf
    80. J. Bellissard, C. Radin and S. Shlosman, The characterization of ground states, J. Phys. A: Math. Theor. 43(2010), 305001. print.pdf
    81. D. Aristoff and C. Radin, Random close packing in a granular model, J. Math. Phys. 51(2010), 113302. print.pdf
    82. H. Koch and C. Radin, Modelling quasicrystals at positive temperature, J. Stat. Phys., 138(2010), 465-475. print.pdf
    83. D. Aristoff and C. Radin, Layering in crumpled sheets, EPL 91(2010), 56003. print.pdf
    84. D. Aristoff and C. Radin, Dilatancy transition in a granular model, J. Stat. Phys. 143(2011), 215-225. print.pdf
    85. T. Argo, M. Guild, P. Wilson, M. Schröter, C. Radin and H. Swinney, Sound speed in water-saturated glass beads as a function of frequency and porosity, J. Acoust. Soc. Am. 129(2011) EL101-EL107. print.pdf
    86. J.-F. Métayer, D.J. Suntrup III, C. Radin, H.L. Swinney and M. Schröter, Shearing of frictional sphere packings, EPL 93(2011), 64003 print.pdf
    87. D. Aristoff and C. Radin, First order phase transition of a long polymer chain, J. Phys. A: Math. Theor. 44(2011), 065004. print.pdf
    88. D. Aristoff and C. Radin, First order phase transition in a model of quasicrystals, J. Phys. A: Math. Theor. 44(2011), 255001. print.pdf
    89. D. Aristoff and C. Radin, Rigidity in solids, J. Stat. Phys. 144(2011) 1247-1255. print.pdf
    90. C. Radin and M. Yin, Phase transitions in exponential random graphs, Ann. Appl. Probab. 23(2013) 2458-2471. print.pdf
    91. D. Aristoff and C. Radin, Emergent structures in large networks, J. Appl. Probab. 50(2013) 883-888. print.pdf
    92. C. Radin, A revolutionary material, Notices Amer. Math. Soc. 60(2013), 310-315. print.pdf
    93. C. Radin and L. Sadun, Phase transitions in a complex network, J. Phys. A: Math. Theor. 46(2013) 305002 print.pdf;    The previous version, with title `A mean field analysis of the fluid/solid phase transition', is here
    94. C. Radin and L. Sadun, Singularities in the entropy of asymptotically large simple graphs, J. Stat. Phys. 158(2015) 853-865 print.pdf
    95. C. Radin, K. Ren and L. Sadun, The asymptotics of large constrained graphs, J. Phys. A: Math. Theor. 47(2014) 175001 print-color.pdf    print-grayscale.pdf
    96. R. Kenyon, C. Radin, K. Ren and L. Sadun, Multipodal structure and phase transitions in large constrained graphs, J. Stat. Phys. 168(2017) 233-258. print
    97. R. Kenyon, D. Král’, C. Radin and P. Winkler, Permutations with fixed pattern densities, Random Structures Algorithms 56(2020) 220-250. print.
    98. R. Kenyon, C. Radin, K. Ren and L. Sadun, Bipodal structure in oversaturated random graphs, Int. Math. Res. Notices 2018(2016) 1009-1044. print
    99. C. Radin, Phases in large combinatorial systems, Ann. Inst. H. Poincaré D 5(2018) 287-308. print.
    100. C. Radin, K. Ren and L. Sadun, A symmetry breaking transition in the edge/triangle network model, Ann. Inst. H. Poincaré D 5(2018) 251-286. print.
    101. C. Radin, The open mathematics of crystallization, Notices Amer. Math. Soc. 64(2017) 551-556. print. There is a more technical version: `Wulff shape for equilibrium phases', arXiv:1610.08564v2 , here
    102. R. Kenyon, C. Radin, K. Ren and L. Sadun, The phases of large networks with edge and triangle constraints, J. Phys. A: Math. Theor. 50 (2017) 435001. print
    103. F. Rietz, C. Radin, H. Swinney and M. Schröter, Nucleation in sheared granular matter, Phys. Rev. Lett. 120(2018) 055701. print.    Supplemental Material.    Movie from Supplementary Material.
    104. C. Radin, K. Ren and L. Sadun, Surface effects in dense random graphs with sharp edge constraint, arXiv:1709.01036v2 preprint
    105. C. Radin and H. Swinney, Phases of granular matter, J. Stat. Phys. 175(2019) 542-553 doi.org/10.10/s10955-018-2144-4 print
    106. J. Neeman, C. Radin and L. Sadun, Phase transitions in finite random networks, J Stat Phys 181 (2020) 305-328, print.    Title of previous version: `Nucleation during phase transitions in random networks'.    Supplementary movies at https://www.youtube.com/playlist?list=PLZcI2rZdDGQqx3WEY6BXoJXcxqQXF5ZzQ
    107. W. Jin, C. O'Hern, C. Radin, M. Shattuck and H.L. Swinney, Homogeneous crystallization in cyclically sheared frictionless grains, Phys. Rev. Lett. 125 (2020) 258003. print.    Supplemental Material.
    108. C. Radin, Conway and aperiodic tilings, Math. Intell. 43 (2021) 15-20, print.
    109. J. Neeman, C. Radin and L. Sadun, Moderate deviations in cycle count, Random Struct. Algorithms (2023), 1-42, https://doi.org/10.1002/rsa.21147, arxiv:2101.08249, preprint.
    110. C. Radin, Equivalence between random close packing in granular matter and freezing in the hard sphere model, arxiv:2105.09982, preprint.
    111. J. Neeman, C. Radin and L. Sadun, Typical large graphs with given edge and triangle densities, Probab. Theory Relat. Fields (2023), https://doi.org/10.1007/s00440-023-01187-8. arxiv:2110.14052.
    112. J. Neeman, C. Radin and L. Sadun, Existence of a symmetric bipodal phase in the edge-triangle model, arxiv:2211.10498, preprint.
    113. C. Radin and L. Sadun, Optimal graphons in the edge-2star model, arxiv:2305.00333 preprint.

    WEB PAGES

    Definitions of thermodynamic phases and phase transitions (AIM workshop, 2006; with M.E. Fisher)

    Quasicrystals: solids with unusual symmetry (Scholarpedia)

    Phase transitions in crumpled matter (IOP Insights)

    Sphere packing and phases

    Complex networks and phases
    VIDEOS

    Lecture to undergraduates (2010): "Water vs. ice"

    Lecture at Vanderbilt conference (2010): "The Stability of Optimizers of Energy or Density"

    Lecture at Newton Institute conference on random graphs, 2016: "Emergent Phases"
    BOOKS

    M. Bowick, D. Kinderlehrer, G. Menon and C. Radin, eds., Mathematics and Materials, IAS/Park City Mathematics Series, Vol 23, Amer. Math. Soc., Providence, 2017

    C. Radin, Miles of Tiles, Student Mathematical Library, Vol 1 (Amer. Math. Soc., Providence, 1999).